ft*.  5- 


Gbe  TUniversitE  of  Cbicacjo 

FOUNDED  BY  JOHN  D.  ROCKEFELLER 

PLACE-CONSTANTS  FOR  ASTER 
PRENANTHOIDES 


SUBMITTED  TO  THE 

SCIENCE, 

A  DISSERTATION 

FACULTY  OF  THE  OGDEN  GRADUATE  SCHOOL  OF 

IN  CANDIDACY  FOR  THE  DEGREE  OF 

DOCTOR  OF  PHILOSOPHY 

(department  of  botany) 

BY 

GEORGE  HARRISON  SHULL 


CHICAGO 

19°4 


Digitized  by  the  Internet  Archive 
in  2017  with  funding  from 

University  of  Illinois  Urbana-Champaign  Alternates 


https://archive.org/details/placeconstantsfoOOshul 


05  C*ki 


PLACE-CONSTANTS  FOR  ASTER  PRENANTHOIDES. 

CONTRIBUTIONS  FROM  THE  HULL  BOTANICAL  LABORATORY. 

LXIV. 

George  Harrison  Shull. 

(with  EIGHTEEN  figures) 

I.  INTRODUCTION. 

Geographic  isolation  has  long  been  accredited  as  an  important 
factor  in  the  process  of  evolution,  but  with  the  introduction  of  methods 
calculated  to  demonstrate  the  evolutionary  processes  an  altogether 
new  conception  has  been  gained  regarding  the  importance  of  locality 
as  a  modifying  factor.  Statistical  methods  have  shown  that  the  organ¬ 
isms  of  any  species  from  different  stations,  often  quite  near  each  other, 
are  not  to  be  considered  homogeneous,  and  that  in  order  to  establish 
a  proper  basis  for  comparison  investigations  must  deal  with  definite 
areas.  The  modal  condition  of  any  species  prevailing  on  such  a 
limited  area  is  known  as  a  “ place-mode”  for  that  species  at  that 
place. 

The  importance  of  determining  and  recording  place-modes  for 
various  species  was  first  emphasized  by  Davenport  (1899a),  and 
in  response  to  his  appeal  a  considerable  number  of  local  statistical 
studies  have  been  made.  Some  of  these  studies  have  shown  that 
the  determination  of  place- constants  is  not  so  simple  a  problem  as 
was  at  first  supposed.  As  a  result  of  my  earlier  studies  on  Aster 
'  (Shull  1902)  it  was  shown  that  the  establishment  of  place-constants 
^  for  a  species  of  Aster  would  involve  the  collection  of  all  the  heads 
produced  during  the  season,  since  there  is  a  continuous  and  more 
or  less  regular  change  in  the  variable  characters  from  day  to  day 
throughout  the  season.  It  was  suggested  there  that  considerable 
0  differences  might  also  be  presented  by  the  same  population  from 
season  to  season.  The  results  of  a  number  of  studies  on  various 
species  by  other  investigators,  both  before  and  since  my  study  of 
Aster,  either  lead  to  the  same  conclusion  or  admit  of  the  same  explana¬ 
tion  (Burkill  1895,  MacLeod  1899,  Ludwig  1901,  Tower  1902, 
Yule  1902,  Pearson  1903,  Reinohl  1903,  etc.). 

1904] 


333 


334 


BOTANICAL  GAZETTE 


[NOVEMBER 


Tower  (1902)  discusses  the  bearing  of  these  results  upon  the 
establishment  of  place-constants,  and  concludes  “that  the  ‘place¬ 
mode  ’  for  a  species  or  for  a  character  of  a  species  should  represent  the 
average  prevailing  condition  at  a  given  place  during  a  period  of 
observation  continued  through  years  or  long  enough  to  eliminate 
the  effect  of  secular  fluctuations. 5,1 

It  has  been  proved  conclusively  that  conditions  of  variability 
which  are  a  function  of  place  are  masked  by.  others  associated  with 
time,  and  before  we  can  satisfactorily  arrive  at  the  one  the  other 
must  be  eliminated.  In  the  efforts  which  have  thus  far  been  made 
to  establish  place- constants  this  fact  has  not  been  taken  into  account. 
Indeed,  we  do  not  yet  know  how  to  take  it  into  account,  since  no 
adequate  investigation  has  been  made  of  changes  in  variability 
which  take  place  during  the  season  and  from  season  to  season.  It 
was  to  add  to  our  knowledge  of  such  secular  variation  and  to  con¬ 
tribute  by  its  elimination  to  the  establishment  of  true  place-constants 
that  the  present  study  was  undertaken. 

1  In  his  summary  Tower  gives  the  following  definition:  “A  ‘place-mode’  is 
the  average  prevailing  state  of  a  homogeneous  lot  of  individuals  \i.  e.,  of  the  same 
pleomorphic  condition  and  stage  of  development]  characteristic  of  a  particular  place 
and  season,  as  determined  by  observations  carried  on  long  enough  to  eliminate  the 
effects  of  secular  climatic  fluctuations.”  The  limitation  of  a  place-mode  to  a  particular 
season  was  plainly  unintentional,  as  it  is  inconsistent  with  the  requirement  that  the 
observations  be  carried  on  long  enough  to  eliminate  the  effects  of  secular  climatic 
fluctuations. 

Pearson  (1902)  objects  to  this  definition  as  not  being  biometric.  He  says: 
“It  might  refer  to  any  constant  whatever  of  the  frequency — to  the  mean,  the  mode, 
the  variability,  or  indeed  to  the  whole  frequency  distribution  itself.”  Tower’s  appli¬ 
cation  of  the  term  place-mode  to  the  average  prevailing  condition  of  a  homogeneous 
population  is  in  harmony  with  a  well-known  philological  principle  which  has  wrested 
very  many  words  from  their  original  signification  to  meanings  of  greater  or  less  inclu¬ 
siveness,  because  the  original  concept  as  limited  by  the  literal  meaning  of  the  symbol 
used  proved  to  be  of  comparatively  little  importance.  In  very  many  instances  the 
mode  is  the  least  important  constant  involved,  and  in  others — particularly  in  the  varia¬ 
tion  of  plants — the  theoretical  mode  is  at  present  indeterminable.  As  no  one  thought, 
in  studying  place-modes,  of  limiting  his  studies  to  the  theoretical  mode,  it  was  not 
unnatural  so  to  extend  the  meaning  of  place-mode  as  to  involve  all  the  quantitative 
relations  of  a  population. 

While  it  was  evidently  Davenport’s  intention  in  proposing  the  word  “place¬ 
mode”  to  use  it  in  its  strict  mathematical  sense,  a  reference  to  his  definition  will  show 
how  easy  it  was  to  make  it  include  the  entire  condition  of  the  population.  He  says 
(1899a):  “I  use  the  word  ‘place-mode’  to  embody  a  well-known  idea,  namely  that  a 


I9°4] 


SHULL— PLACE-CONSTANTS  FOR  ASTER 


335 


II.  MATERIAL  AND  METHODS. 

This  study  is  based  upon  the  flowering  heads  of  Aster  prenan- 
thoides  Muhl.  collected  thrice  a  week  from  the  same  area  which 
furnished  the  serial  collections  for  my  earlier  studies  (Shull  1902). 
This  species  is  in  some  ways  an  ideal  subject  for  studies  of  this  kind. 
The  heads  are  beautifully  regular,  as  may  be  seen  in  fig.  1.  They 
are  little  subject  to  injuries,  and  almost  the  only  heads  which  must 
be  thrown  away  are  those 
in  which  insect  larvae  have 
hatched  and  subsisted  upon  the 
developing  flowers.  Such  cases 
are  not  numerous,  amounting 
to  less  than  2  per  cent,  of  the 
heads  collected  this  year. 

The  personal  equation  was 
eliminated  in  the  same  manner 
as  in  my  former  work,  i.  e.,  by 
the  collection  of  every  head 
that  bloomed  on  a  naturally  cir¬ 
cumscribed  area.  The  method 
used  in  making  the  counts  was 
also  the  same,  the  heads  being 
completely  dissected.  This 
method  prevents  the  errors 
which  will  frequently  occur  in 
the  counting  of  rays  without 
Complete  dissection  errors  jrIG  x — Aster  prenanthoides  at  Clifton,  Ohio. 

s  pecies  has  a  different  mode  ( i ,  e.,  a  different  prevailing  condition  of  size,  color,  etc. 
in  different  localities.  The  person  who  seeks  to  determine  a  place-mode  determines 
the  prevailing  dimension  of  the  principal  measurable  qualities  (and  practically  all 
qualities  of  organisms  are  measurable)  of  a  species  as  it  occurs  in  the  locality  in  question.” 

It  seems  desirable  in  applying  the  exact  methods  of  the  mathematician  to  biology 
to  retain  the  exact  mathematical  significance  of  the  word  “  place-mode,”  though  the 
term  then  becomes  of  comparatively  little  usefulness.  As  a  more  inclusive  term, 
involv/ng  the  characteristics  of  any  or  all  the  measurable  qualities  of  a  species  a 
represented  by  a  lot  of  individuals  occurring  at  any  place  in  question,  let  us  adopt  the 
word  place-condition  or  place-habit.  We  may  then  say  that  we  investigate  the  place- 
habit  or  place-condition  of  a  species  at  a  particular  place  in  order  to  determine  or 
establish  the  place-constants,  among  these  being  the  place-mode. 


336 


BOTANICAL  GAZETTE 


[NOVEMBER 


resulting  from  the  loss  of  rays  due  to  age  or  other  causes — and  it  saves 
the  necessity  of  discarding  any  material  on  this  ground,  since  the 
remains  of  the  ray-flowers  are  always  distinguishable  from  the  disk- 
flowers  when  they  are  separated. 

It  ought  not  to  be  necessary  in  work  of  this  kind  to  give  assurance 
that  no  material  has  been  arbitrarily  discarded,  either  in  collection 
or  in  seriation,  but  the  importance  of  this  matter  seems  to  be  too 
little  appreciated.  If  one  student  arbitrarily  discards  material, 
who  else  in  working  over  the  same  material  will  arbitrarily  discard 
on  the  same  basis  ?  And  if  an  investigation  cannot  be  repeated 
by  another  investigator  with  at  least  approximately  identical  results, 
of  what  value  is  it?  Tower  (1902)  assumed  that  his  failure  to  get 
a  mode  at  34  in  Chrysanthemum  Leucanthemum  might  be  due  to  the 
fact  that  he  discarded  a  number  of  heads  on  account  of  age.  He 
states  that  counts  of  some  of  this  rejected  material  showed  that  all 
of  the  heads  had  a  large  number  of  rays.  What  would  have  been 
the  result  had  he  counted  all  the  heads  he  rejected?  Miss  Small¬ 
wood  (1903)  “arbitrarily  threw  out  the  small”  specimens  of  beach-flea 
and  then  presented  statistics  as  to  the  size  and  variability  of  the 
remainder,  as  if  these  data  could  have  either  interest  or  scientific 
value.2  If  anomalies  appear  when  all  the  data  are  seriated,  they 
should  be  explained  if  possible,  but  explained  or  unexplained  the 
data  should  be  given,  because  these  have  value  whether  the  explana¬ 
tion  has  or  not. 

In  calculating  the  various  constants  I  have  again  used  the  formulae 
tabulated  in  Davenport’s  (18996)  Statistical  methods ,  except  that 
instead  of  Duncker’s  method  of  calculating  the  coefficient  of  corre¬ 
lation  I  have  used  the  neat  method  adopted  by  Yule  (1897),  which 
may  be  expressed  by  the  formula 


(%jx'x"  ,  1 

={—t - V‘  V- 


in  which  x'  and  x"  are  the  deviations  from  an  integral  assumed 
mean  of  subject  and  relative  classes  respectively,  /  is  the  frequency 
of  occurrence  of  each  combination  of  subject  and  relative  deviations, 

2  In  justice  to  Miss  Smallwood  it  should  be  said  that  her  paper  deals  chiefly 
with  the  ethological  relations  of  the  beach-flea,  and  that  she  seems  to  have  appreciated 
the  unsatisfactory  character  of  her  quantitative  results. 


19°4] 


SHULL— PLACE-CONSTANTS  FOR  ASTER 


337 


n  is  the  whole  number  of  variates,  v/  and  vt”  are  the  deviations  of 
the  assumed  means  from  the  true  means,  and  a'  and  a"  the  errors 
of  mean  square  or  “standard  deviations”  respectively  of  the  subject 
and  relative  categories.3 

III.  LOCALITY  AND  HABITAT. 

Flowers  collected  from  the  hillsides  differ  in  a  marked  way  from 
those  of  the  same  species  collected  in  the  lowlands  of  the  same  locality, 
as  has  been  shown  in  many  instances  by  de  Vries,  Ludwig,  Reinohl, 
and  others.  This  is  a  question  of  habitat.  It  remains  an  unsolved 
problem  whether  plants  are  not  so  sensitive  to  edaphic  and  local 
climatic  conditions  as  to  make  impossible  the  derivation  by  statistical 
methods  of  anything  more  fundamental  than  the  fact  and  the  degree 
of  this  extreme  sensitiveness.  This  problem  can  be  solved  only  by 
long  and  carefully  conducted  investigations.  In  order  that  we  may 
discriminate  between  the  influences  of  habitat  and  locality  in  making 
studies  in  variation,  it  becomes  necessary  to  record  as  carefully  as 
possible  the  habitat  in  which  the  material  has  been  collected,  and  so 
to  indicate  the  locality  that  future  investigators  may  visit  the  identical 
area  studied. 

The  definite  character  both  of  habitat  and  of  locality  has  strongly 
commended  the  choice  of  this  particular  area  of  Aster  prenanthoides 
for  such  thorough  investigation  as  is  needed  to  elucidate  the  complex 
problems  involved  in  work  of  this  kind. 

Clifton  is  a  small  village  on  the  boundary  between  Clark  and 
Greene  counties,  Ohio,  in  lat.  390  48'  43"  north  and  long.  83°  48'  41" 
west.  The  Little  Miami  River,  on  whose  northern  bank  the  village 
lies,  occupies  a  post-glacial  channel  in  massive  gray  Niagara  lime¬ 
stone,  forming  a  deep  and  narrow  gorge  widely  known  for  the  beauty 
of  its  scenery.  About  one  kilometer  west  of  the  village  two  small 
streams  enter  the  river  from  the  north.  Both  of  these  tributary 
streams  have  cut  gorges  in  the  limestone,  but,  being  unable  to  corrade 
their  beds  so  rapidly  as  does  the  river,  they  have  been  left  high  above 
the  level  of  the  river  in  hanging  valleys.  At  about  10-20  m  from  the 
river  the  Yellow  Springs  turnpike  crosses  these  streams  by  two  stone 
arches.  The  area  chosen  as  the  source  of  material  for  this  study  is 

3  This  method  displaces  Duncker’s  method  in  the  second  edition  of  Davenport’s 
Statistical  methods  which  has  just  appeared. 


338 


BOTANICAL  GAZETTE 


[NOVEMBER 


that  portion  of  the  more  westerly  tributary  ravine  lying  between 
the  road  and  the  precipice  over  which  the  stream  falls  into  the  river. 
The  locality  is  indicated  by  a  star  on  the  map  (fig.  2). 

An  important  theoretical  consideration  is  the  relation  of  the  locality 
to  the  whole  range  of  the  species  and  to  the  direction  of  its  migration. 

What  relation  does  the  vari¬ 
ability  of  a  species  near  its 
limit  of  range  bear  to  that 
at  its  center  ?  Is  there  a  pro¬ 
gressive  change  of  variable 
characters  along  the  lines  of 
migration  radiating  from  the 
center  of  distribution  of  the 
species?  Adams  (1902) 
accepts  such  progressive 
change  as  one  of  his  ten 
criteria  for  the  determina- 

Fig.  2. — Map  of  Clifton  and  vicinity;  station  . 

for  Aster  prenanthoides  marked  with  a  star.  ^lon  centers  of  distribu¬ 

tion.  He  concludes  that  the 
southeastern  United  States  has  been  the  principal  center  of  post-glacial 
distribution  for  the  eastern  half  of  North  America.  The  determination 
of  place-constants  in  various  parts  of  the  range  would  test  the  value  of 
this  criterion.  It  is  to  be  hoped  that  investigators  in  other  places  will 
make  studies  similar  to  this  for  the  purpose  of  throwing  a  more  certain 
light  on  these  principles. 

To  show  the  relation  of  Clifton,  Ohio,  to  the  total  range  of  Aster 
prenanthoides  I  present  in  fig.  j  a  map  showing  the  range  of  the 
species  as  represented  at  the  present  time  by  specimens  in  the  leading 
American  herbaria.  Such  a  map  is  always  to  some  extent  a  commen¬ 
tary  on  the  limitations  of  the  herbaria,  and  does  not  correctly  repre¬ 
sent  the  relative  frequency  of  the  species  in  the  different  parts  of  its 
range.  E.  S.  Burgess,  the  well-known  specialist  on  the  genus  Aster, 
writes  that  the  stations  in  the  Berkeley  Hills,  Mass.,  and  in  the 
Catskills  and  Shawangunk  Mountains,  N.  Y.,  are  really  extra- 
limital,  “the  species  becoming  common  300  miles  [480  km]  westward, 
and  reaching  proper  development  along  the  southern  shore  of  Lake 
Erie  and  thence  through  western  Pennsylvania.”  The  fewness  of 


i9°4] 


SHULL— PLACE-CONSTANTS  FOR  ASTER 


339 


the  stations  in 'the  western  half  of  the  range  is  due  to  the  general 
scarcity  of  herbarium  material  from  this  area  rather  than  to  the 
rarity  of  the  species.  It  will  be  seen  from  this  map  that  the  Clifton 
station  (marked  with  a  star)  is  very  nearly  central. 


Fig.  3. — The  range  of  Aster  prenanthoides  as  represented  by  specimens  in  the 
leading  American  herbaria ;4  Clifton,  Ohio,  marked  with  a  star. 


4  The  stations  represented  on  the  map  are  as  follows:  Massachusetts:  Berkeley 
Hills.  New  York:  Chenango  Co.,  Oswego  Co.,  Tompkins  Co.,  Andover,  Big  Tree, 
Bridgewater,  Buffalo,  Elmira,  Macedon,  Catskill  Mts.,  Shawangunk  Mts.  New 
Jersey:  Woodbury.  Pennsylvania:  Bedford  Co.,  Chester  Co.,  Conewago  Co., 
Lancaster  Co.,  Westmoreland  Co.,  Easton,  Mercersburg,  Philadelphia,  Trout  Run. 
Maryland:  Baltimore,  Emmittsburg.  District  of  Columbia:  Washington.  Vir¬ 
ginia:  Big  Stone  Gap,  Pulaski,  Wytheville.  West  Virginia:  Aurora.  Kentucky: 


340 


BOTANICAL  GAZETTE 


[NOVEMBER 


The  habitat  of  the  population  in  question  is  typical  of  the  species, 
if  we  may  judge  from  the  statements  made  in  the  various  manuals,  all 
of  which  agree  that  the  characteristic  habitat  is  along  margins  of 
streams  in  shady  places.  This  fact  will  make  it  easy  to  find  these 
plants  growing  under  essentially  the  same  conditions  in  other  localities, 
and  thus  facilitate  the  establishment  of  place-constants  which  shall  be 


Fig.  4. — The  habitat,  looking  south. 


properly  comparable.  The  general  character  of  the  habitat  at  Clifton 
will  be  best  understood  by  a  reference  to  jigs.  1,  4 ,  and  5.  It  will  be  at 
once  recognized  that  we  have  here  an  example  of  temporary  mesophy- 
tic  climax  characteristic  of  young  ravines,  the  luxuriance  of  vegeta- 

Bell  Co.,  Carter  Co.,  Lexington.  Ohio:  Franklin  Co.,  Berea,  Cleveland,  Clifton, 
Ironton,  Mansfield,  Wooster.  Indiana:  Hamilton  Co.  Illinois:  Canton.  Michi¬ 
gan:  Allegan  Co.,  Keweenaw  Co.  Wisconsin:  Eau  Galle,  Milwaukee,  Racine, 
Sparta.  Minnesota:  Mazeppa.  Iowa:  Fayette  Co.,  Johnson  Co.,  Ames.  Kansas: 
Neosho  Co. 


1904] 


SHULL— PLACE-CONSTANTS  FOR  ASTER 


341 


tion  being  due  to  the  moderation  of  extremes  of  temperature,  light, 
etc.,  the  protection  from  winds,  and  the  consequent  maintenance  of  a 
relatively  high  degree  of  humidity.  Besides  the  relative  constancy 
of  atmospheric  conditions,  it  should  be  noted  that  the  stream  '^hich 
occupies  the  ravine,  and  along  whose  margin  the  Aster  prenanthoides 
is  growing,  is  a  permanent,  spring-fed  stream  draining  so  small  an 


Fig.  5. — The  habitat,  looking  north. 

area  as  to  be  little  subject  to  fluctuations  due  to  the  alternations  of 
dry  and  rainy  periods.  The  map  shown  in  jig.  6  accurately  represents 
the  present  position  of  the  stream  and  the  location  of  the  areas  of 
Aster  prenanthoides  with  reference  to  it,  thus  enabling  the  future 
investigator  to  note  any  changes  of  area  or  relation  which  might  have 
an  influence  upon  the  variability. 

IV.  RESULTS. 

The  collections  having  been  made  about  three  times  a  week, 
regardless  of  the  number  of  heads  which  were  in  bloom  at  any  time, 
it  does  not  seem  desirable  to  present  curves  and  correlation  tables 


342 


BOTANICAL  GAZETTE 


[NOVEMBER 


FIG.  6. — Map  showing  the  relation  of  Aster  prenanthoides  to  the  stream  at  the 
Clifton  station;  heavy  contours  represent  the  upper  margin  of  the  cliffs;  light  contours 
show  the  limit  of  the  flood  plains;  depth  of  ravine  5-7  m;  height  of  hanging  valley 
above  river  level  7-8  m. 

of  each  separate  collection.  In  order  to  make  the  original  data 
available  for  any  purpose  to  which  other  students  may  wish  to  turn 
it,  I  give  in  tabulated  form  the  results  of  the  several  collections  (Tables 
A,  B,  and  C). 


I9°4] 


SHULL— PLACE-CONSTANTS  FOR  ASTER 


343 


TABLE  A. 

BRACTS. 


344 


BOTANICAL  GAZETTE 


[NOVEMBER 


-Ac¬ 


table  B. 


RAYS. 


TABLE  C. 

DISK-FLORETS. 


i9°4] 


SHULL— PLACE-CONSTANTS  FOR  ASTER 


345 


TABLE  C  ( continued ). 


b 


346 


BOTANICAL  GAZETTE 


[NOVEMBER 


TABLE  D. 

CHIEF  CONSTANTS  OF  THE  SEVERAL  COLLECTIONS. 


Bracts 

Sep.  12 

Sep.  14 

Sep. 16 

Sep. 18 

Sep. 21 

Sep. 23 

Sep. 25 

Sep. 28 

Sep.  30 

Oct.  2 

Oct.  6 

Oct.  9 

Mean . 

36.555 

43-297 

43-174 

43-898 

40 . 700 

40.652 

39.828 

37.423 

36.016 

35.120 

33-096 

35-750 

P.  E.  M.... 

±  1.  no 

±  .868 

±  .947 

±  .500 

±  -723 

±  .492 

±  .598 

±  .401 

±  -543 

±  .423 

±.389 

±  -935 

<T . 

9.870 

7.829 

6-735 

6.951 

8.303 

7.002 

7.090 

6.598 

!  6.438 

5-717 

5-247 

2.773 

P.  E.  a-  . .  . 

±  .785 

±  .614 

±  .670 

±  -353 

±  .511 

±  .348 

±  .423 

±  .  284 

±  .384 

±  .299 

±  -275 

±  .661 

C.  V . 

27.000 

18.081 

15.600 

15-835 

20.401 

17-225 

17.801 

17.630 

17.877 

16.278 

15-854 

7-756 

P.  E.  v  •  •  • 

±2. 146 

± 1.418 

±i-55i 

±  .805 

± 1.256 

±  .857 

± 1 .061 

±.758 

± 1.066 

±  .852 

±  .830 

± 1 . 849 

Rays 


Mean . 

23-305 

27.540 

26.739 

28.681 

26.600 

26.033 

26.187 

24.179 

24.109 

23.880 

22.301 

24.500 

P.E.m-... 

±  .  720 

±  .582 

±  .669 

±  .380 

±  .485 

±  .326 

±  .392 

±  .249 

±  -342 

±  .272 

±  .278 

±  -559 

O' . 

6.407 

5-248 

4-757 

5.284 

5-571 

4.628 

4-650 

4.092 

4-058 

3.668 

3-757 

1.658 

P.  E.  <r  . . . 

±  .509 

±  .412 

±  -473 

±  .269 

±  -343 

±  .  230 

±  .277 

±  .  176 

±  .242 

±  .  192 

±  .197 

±  .396 

C.  V . 

27.490 

19-158 

17.790 

18.423 

20.945 

17.780 

17.756 

16.926 

16.834 

i5-36i 

16 . 846 

6.769 

P.  E.  v _ 

±2.185 

±1.494 

± 1.769 

±  -937 

± 1.290 

±  .884 

±1.059 

±  .  728 

± 1 . 004 

±  .804 

±  .882 

± 1.614 

Disk- florets 


Mean . 

.  E.  m  — 


P.  E.  <r  ... 

C.  V . 

P.  E.  v _ 


41.611 

Si-729 

46.522 

5I.5II 

47 • 100 

46.891 

47.672 

43 • 569 

42.187 

41.301 

38.072 

40.250 

± 1.282 

±1.318 

±1-343 

±  .724 

±  .862 

±  .603 

±  .813 

±  -  459 

±  .682 

±  .469 

±  -535 

± 1 • 103 

11.405 

11.884 

9-546 

10.062 

9.899 

8.569 

9.647 

7-539 

8.087 

6.339 

7.219 

3269 

±  .907 

±  .932 

±  -949 

±  .512 

±  .610 

±  .426 

±  -575 

±  -324 

±  .482 

±  -332 

±  .378 

±  .780 

27.408 

22.973 

20.519 

19-534 

21.017 

18.274 

20.236 

I7-303 

19.170 

15-349 

18.962 

8.122 

±2.179 

± 1 .801 

±  2.041 

±  -993 

±1.294 

±  .909 

± 1 . 206 

±  -744 

±1.143 

±  .804 

±  -993 

±i-937 

In  Table  D  are  given  the  more  important  constants  of  the  several 
collections  with  the  probable  errors  of  the  determination.  The 
average  deviation  is  omitted  as  having  no  significance,  since  the  prob¬ 
able  error  of  the  determination  is  almost  as  large  as  the  determined 
value  of  the  constant.  It  will  be  noted  on  examining  this  table  that 
all  the  constants  are  quite  variable,  and  that  only  the  mean  seems  to 
follow  a  rather  definite  law,  beginning  low,  then  leaping  almost 
immediately  to  the  maximum,  after  which  there  is  a  gradual  fall 
until  almost  the  end,  when  a  slight  rise  appears.  The  fall  of  mean 
values  from  the  maximum  on  September  18  to  the  minimum  on 
October  6  was  24.6  per  cent,  in  bracts,  22.25  Per  cenh  in  rays,  and 
26.4  per  cent,  in  disk-florets.  The  changes  of  mean  value  for  each 
set  of  variants  from  the  beginning  to  the  end  of  the  flowering  season 
and  the  corresponding  changes  observed  in  1 900  are  shown  graphically  . 
in  fig.  7. 


I9°4] 


SHULL— PLACE-CONSTANTS  FOR  ASTER 


347 


i.  Bracts. — The  frequency  polygon  for  the  bracts  is  shown  in 
fig.  8.  The  mean  value  is  38.5971+1. 189,  and  a  number  of  empirical 
modes  are  present.  Some,  if  not  all,  of  these  are  doubtless  due  to 
the  smallness  of  the  number  of  heads  counted.  It  will  be  noted 

September  October 


Fig.  7. — Curves  showing  the  changes  in  the  mean  numbers  of  parts  in  the  heads 
from  the  beginning  to  the  end  of  the  season,  and  the  difference  between  the  two  collec¬ 
tions,  1900  and  1903. 

that  the  most  prominent  modes  are  those  which  lie  below  the  mean 
value;  i.  e.,  if  one  may  speak  of  skewness  in  multimodal  curves,  there 
is  evident  here  a  strong  positive  skewness.  The  breadth  of  the 
curve  exhibits  to  the  eye  the  great  variability,  which  may  be  expressed 


34§ 


BOTANICAL  GAZETTE 


[NOVEMBER 


numerically  by  the  standard  deviation  7.692 ±.133,  or  by  the  coeffi¬ 
cient  of  variability  19.928 ±  .345. 

2.  Rays. — The  ray-curve  shown  in  fig.  9  has  the  mean  at  25.247 
±.122  and  by  much  the  strongest  mode  at  22,  so  that  here  again 
there  is  a  very  prominent  skewing  of  the  curve.  The  empirical 


Fig.  8. — Bract-curve  for  757  heads  collected  in  1903:  mean  38. 597 ±.189;  empir¬ 
ical  modes  29-30,  34/39,  4*,  46,  49,  5U  <r  =  7-692±  -133- 


modes  at  16,  22,  26,  33  are  somewhat  suggestive  of  the  series  which 
Ludwig,  de  Vries,  and  others  have  shown  to  be  so  common,  but 
aside  from  such  suggestion  can  have  little  meaning.  The  standard 
deviation  4.990  ±.087  is  considerably  less  than  that  of  the  bracts, 
while  the  coefficient  of  variability,  19. 764 ±.344,  is  practically  the 


same. 


i9°4] 


SHULL— PLACE-CONSTANTS  FOR  ASTER 


349 


Fig.  9.— Ray-curve  for  757  heads  collected  in  1903:  mean  25. 247 ±  .122;  empir 
ical  modes  16,  22,  26,  33;  <r  =4.9899 ±  .0865. 


35° 


BOTANICAL  GAZETTE 


[NOVEMBER 


3.  Disk- florets. — The  polygon  of  distribution  of  the  disk- florets 
is  presented  in  fig.  10.  The  range,  from  14  to  78,  is  so  wide  that  757 
heads  are  quite  insufficient  to  make  the  many  empirical  modes  of  any 


Fig.  10. — Disk-floret  curve  for  757  heads  collected  in  1903:  classes  doubled; 
mean  44. 933 ±  -2385  empirical  modes  38-39,  44~45>  5°-5x>  56~57>  64-65;  <>-  =  9.703 
±  .168. 

significance.  To  make  the  curve  comparable  with  that  offered  by 
the  disk- florets  in  1900,  the  classes  were  doubled,  but  this  still  leaves 


i9°4] 


SHULL— PLACE-CONSTANTS  FOR  ASTER 


351 


five  prominent  modes.  The  positive  skewness  is  not  so  marked  as 
in  the  bracts  and  rays,  but  inspection  of  the  curve  will  show  plainly 
that  the  theoretical  mode  is  below  the  mean,  though  the  principal 
apparent  mode  nearly  coincides  with  the  mean.  The  standard 
deviation  is  9.703  ±.168  and  the  coefficient  of  variability  21.595  ±. 3 74. 

4.  Summation ,  igoo  and  igoj. — The  more  important  constants 
for  bracts,  rays,  and  disk- florets  are  given  in  Tables  E,  F,  and  G, 
along  with  the  corresponding  data  for  1900  and  the  summation  of 
the  two  lots.  The  summation-curves  of  bracts,  rays,  and  disk- 

TABLE  E. 


CONSTANTS  OF  BRACTS  FOR  19OO,  1903,  AND  THEIR  SUM. 


1900 

1903 

Combined 

Number . 

658 

757 

1415 

Mean . 

44.044 

38-597 

41.130 

P.  E.m . 

±  -i5° 

±  .  189 

±  .131 

f  34,  39, 

f  29-3°>  34, 

f  30,  32,  34, 

Modes,  empirical .... 

\  39,  4i,  46, 

\  39,  4i, 

1 43,  49 

l49,  51 

1 43,  49 

5-7I7 

7.692 

7-3I3 

P.  E.o- . 

±  .  106 

±•133 

±  .093 

C.  V . 

12 -979 

19.928 

1 7  *  775 

P.  E.  v . 

±  .241 

±  -345 

±  .225 

TABLE  F. 

CONSTANTS  OF  RAYS  FOR  I9OO,  1903,  AND  THEIR  SUM. 


1900 

1903 

Combined 

Number . 

658 

757 

1415 

Mean . 

28.038 

25.247 

26.545 

P.  E.m . . . 

±  .  107 

± .  122 

±  .086 

Modes,  empirical. . .  . 

27,  3L  33 

16, 22, 26, 33 

16,  22,  26,  31,  33 

<T . 

4.070 

4.990 

4.792 

P.  E.  o- . 

±  .076 

±  .087 

±  .061 

C.  V . 

14.516 

19.764 

18.052 

P.E.y . 

H- 

10 

O 

±  -344 

±  .229 

florets,  together  with  the  two  partial  curves  of  which  each  is  com¬ 
posed,  are  shown  in  figs,  u,  12,  and  13.  These  curves  are  all  reduced 
to  the  same  area,  500  units,  in  order  to  facilitate  comparison,  and 
the  method  of  “loaded  ordinates’’  is  used  to  allow  the  curves  to  be 
superposed. 


352 


BOTANICAL  GAZETTE 


[NOVEMBER 


The  bracts  for  the  two  years  combined  present  no  less  than  seven 
empirical  modes,  showing  without  question  that  1415  heads  is  still 
too  few  for  material  having  so  wide  a  range  and  so  high  standard 
deviation.  In  the  rays,  the  range  being  less  and  the  standard  devia- 


Fig.  11. — Summation  curve  for  bracts  1900  +  1903  and  the  bract-curves  for  1900 
and  1903  superposed  for  comparison;  all  reduced  to  the  same  area,  500  units;  heavy 
line  the  summation  curve;  dotted  line  the  1900  curve;  dot  and  dash  line  the  1903  curve. 


TABLE  G. 


CONSTANTS  OF  DISK-FLORETS  FOR  I9OO,  1903,  AND  THEIR  SUM. 


1900 

1903 

Combined 

Number . 

658 

757 

1415 

Mean . 

50.298 

44-933 

47.428 

P-  E.  M . 

£_±  .  166 

r  j  48-49? 

! 

±  .238 

r 38-39? 
44-45? 

±  -156 

Modes,  empirical .... 

il  52-53 

r 

1 

1 

1 

50-51? 

56-57? 

.  64-65 

49-50 

r  6.310 

' 

9- 7°3 

8.724 

P.  E.o- . 

± .  117 

±  .168 

±  .ill 

C.  V . . 

12.546 

21-595 

18.395 

P.  E.  v . 

±  •  233 

±  -374 

±  -233 

i9°4] 


SHULL— PLACE-CONSTANTS  FOR  ASTER 


353 


tion  only  two-thirds  as  great,  the  number  of  heads  is  much  more 
nearly  adequate.  There  are  presented  five  empirical  modes,  16,  22, 
26,  31,  33,  almost  in  agreement  with  Ludwig’s  series. 

5.  Correlations. — The  correlation  between  rays  and  bracts  is 
shown  in  fig.  14,  between  rays  and  disk- florets  in  fig.  15,  and  between 


Fig.  12. — Summation  curve  for  rays  1900  +  1903  and  the  ray-curves  for  1900  and 
1903  superposed  for  comparison;  all  reduced  to  the  same  area,  500  units;  heavy  line 
the  summation  curve;  dotted  line  the  1900  curve;  dot  and  dash  line  the  1903  curve. 

bracts  and  disk-florets  in  fig.  16.  The  coefficient  of  correlation  is 
very  high  in  all,  being  greatest  between  rays  and  bracts,  and  least 
between  rays  and  disk-florets.  The  coefficients  may  be  compared 
with  those  of  1900  in  Table  H: 


354 


BOTANICAL  GAZETTE 


[NOVEMBER 


Fig.  13. — Summation  curve  for  disk-florets  1900+  1903  and  the  disk-floret  curves  for  1900  and  1903  superposed  for 
comparison;  all  reduced  to  the  same  area,  500  units;  heavy  line  the  summation  curve;  dotted  line  the  1900  curve;  dot  and 
dash  line  the  1903  curve. 


i9°4] 


SHULL— PLACE-CONSTANTS  FOR  ASTER 


355 


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Fig.  14. — Correlation  surface  for  757  heads  collected  in  1903;  rays  subject,  bracts  relative;  p=  .8745±  .0042. 


356 


BOTANICAL  GAZETTE 


[NOVEMBER 


TABLE  H. 


COEFFICIENTS  OF  CORRELATION. 


1900 

1903 

Rays  and  bracts . 

. 7051 ±  .0092 
.6749±  .0100 

.8745±  .0042 
.8240±  .0058 
.8355±  .0055 

Rays  and  disk-florets . 

Bracts  and  disk-florets . 

V.  DISCUSSION. 

In  my  earlier  study  the  conclusion  was  reached  that  the  mean 
number  of  parts  in  the  heads  of  Aster  prenanthoides  begins  high 
and  falls  continuously  from  the  beginning  to  the  end  of  the  flowering 
season.  This  was  recognized  as  in  accord  with  Burkill’s  (1895) 
results  on  Alsine  media  and  other  species.  Reinohl  (1903)  has 
recently  made  a  very  careful  study  of  Alsine  media  and  reports  that 
the  first  flowers  never  have  the  highest  number  of  stamens  and  that 
the  maximum  number  is  reached  only  after  some  time.  He  attributes 
Burkill’s  results  to  the  fact  that  they  were  based  upon  occasional 
collections  which,  he  supposes,  did  not  happen  to  involve  the  very 
earliest  flowers  of  the  season.  My  collections  of  Aster  prenanthoides 
in  1900  included  all  the  heads  which  bloomed  that  year,  but  the  first 
collection  was  made  so  late  that  the  mean  numbers  of  parts  in  the 
very  earliest  heads  were  indeterminable  because  of  their  association 
with  heads  of  later  development;  but  in  1903  the  collections  were 
made  with  such  frequency  as  more  closely  to  analyze  the  changes 
taking  place  during  the  season,  there  being  presented  here  twelve 
successive  collections  instead  of  four. 

It  is  now  shown  that  in  Aster  prenanthoides  also  the  mean  numbers 
begin  low,  leaping  almost  immediately  to  the  maximum,  and  thence 
falling  more  gradually  till  near  the  end  of  the  season.  Inspection  of 
fig.  7  will  make  it  clear  that  four  collections  in  1903,  made  on  the 
same  dates  as  the  1900  collections,  would  have  led  to  the  conclusions 
then  reached  that  there  is  a  continuous  fall  from  the  beginning  to  the 
end  of  the  season. 

It  will  be  noted  in  the  same  figure  that  the  last  collection  in  1903 
shows  a  rise  in  mean  values.  As  this  collection  consisted  of  but  four 
heads,  it  can  be  considered  as  having  little  significance.  I  believe, 
however,  that  this  condition  will  be  found  to  occur  not  infrequently. 


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Fig.  15. — Correlation  surface  for  757  heads  collected  in  1903;  rays  subject,  disk-florets  relative;  .  8240 ±  .0058. 


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Fig.  16. — Correlation  surface  for  757  heads  collected  in  1903;  bracts  subject,  disk-florets  relative;  p  — .8355  ± 


i9°4] 


SHULL— PLACE-CONSTANTS  FOR  ASTER 


357 


It  was  observed  by  Burkill  (1902)  in  Ranunculus  arvensis  L.,  but 
he  did  not  “feel  justified  in  suggesting  a  cause  for  it.”  Burkill 
(1895)  showed  in  his  earlier  paper  that  in  Caltha  palustris,  Ranun¬ 
culus  arvensis ,  and  R.  bulbosus  the  first  flower  of  any  individual  has 
a  higher  number  of  stamens  than  any  subsequent  flower  of  that 
individual.  Haacke  (1896)  points  out  the  same  fact  in  regard  to 
the  number  of  rays  in  the  heads  of  Tanacetum  ( Chrysanthemum ) 
corymbosum.  If  this  is  also  true  in  Aster  prenanthoides  (and  I  believe 
it  is),  how  are  we  to  account  for  the  peculiarities  of  the  curves  in  fig.'/ .? 
The  general  fall  in  mean  values  from  near  the  beginning  to  near  the 
end  of  the  flowering  period  can  be  best  explained  perhaps  by  the 
gradual  waning  of  vegetative  vigor  during  the  time  at  which  the 
differentiation  takes  place  which  determines  the  number  of  parts 
in  the  heads.  This  decreasing  vigor  was  supposed  by  Burkill  (1895) 
to  be  largely  due  to  changing  temperature,  but  Reinohl  (1903)  has 
shown  that  temperature  has  little  if  any  influence,  while  the  important 
factor  seems  to  him  to  be  that  of  available  food-supply.  It  is  conceiv¬ 
able  that  there  may  be  a  decreasing  lability  of  the  protoplasm  result¬ 
ing  from  lessened  water-supply,  or  the  accumulation  of  inert  products 
of  metabolism,  or  from  other  causes,  which  would  bring  about  a 
progressive  fall  in  the  number  of  parts  in  the  heads,  even  though  the 
food- supply  remained  unchanged. 

But  if  every  individual  produces  the  highest  number  of  parts  in  the 
first  head  that  blooms  and  the  lowest  number  of  parts  in  the  . last,  how 
can  the  mean  number  begin  low — far  below  the  maximum — and  end 
with  a  rise  ?  This  is  to  be  explained  by  the  fact  that  we  are  dealing 
with  a  population  instead  of  an  individual.  The  precocious  flowering 
of  starved  or  otherwise  weakened  individuals  is  a  well-known  phe¬ 
nomenon,  and  it  is  evident  that  the  heads  gathered  at  the  first  collection 
were  those  produced  by  the  very  weakest  individuals,  and  owe  their 
low  values  to  that  fact.  Very  soon,  however,  the  mediocre  plants, 
composing  the  great  majority  of  the  population,  begin  to  bloom, 
thus  bringing  the  mean  values  at  once  to  the  maximum,  from  which 
they  gradually  fall  until  almost  the  end  of  the  season.  The  very 
last  to  bloom  will  undoubtedly  be  the  last  heads  of  certain  very 
vigorous  individuals  which  did  not  begin  to  bloom  till  late  in  the  season, 
and  though  these  heads  have  the  lowest  numbers  of  parts  produced 


358 


BOTANICAL  GAZETTE 


[NOVEMBER 


by  those  individuals,  they  yet  have  higher  numbers  than  the  last 
heads  of  the  mediocre  portion  of  the  population  which  determined 
the  minimum  mean  value.  These  facts  will  be  made  clear  by  a 
reference  to  fig.  iy,  in  which  the  numerous  oblique  parallel  lines 
represent  the  change  in  mean  number  of  parts  in  the  heads  of  the 

individual  stems  com¬ 
posing  the  population. 
The  abscissal  distances 
of  the  ends  of  a  given 
line  indicate  the  time 
at  which  the  individ¬ 
ual  represented  by  that 
line  began  to  bloom 
and  that  at  which  it 
ceased  blooming,  while 
the  ordinatal  distances 
of  the  same  points  rep¬ 
resent  the  number  of 
parts  in  the  first  and 
last  heads  produced. 
The  heavy  line  running 
through  the  middle  of 
the  figure  is  the  exact 
mean  of  the  figure  as 
drawn.  This  figure  is 
somewhat  schematic, 
of  course,  but  it  is  not 
wholly  imaginary.  The 
mean  is  suggested  by  the  1903  bract-curve  of  fig.  y,  and  the  outline 
will  be  recognized  in  the  distribution  of  numbers  in  Table  A,  which 
also  belongs  to  the  bracts  of  the  1 903  -  population,  so  that  fig.  iy 
may  be  looked  upon  as  a  slightly  schematized  representation  of  the 
bracts  as  they  actually  occurred  in  1903. 

In  1900  the  change  in  mean  values  from  the  maximum  to  the 
minimum  was  11.6  per  cent,  in  bracts,  14.4  per  cent,  in  rays,  and 
18.9  per  cent,  in  disk-florets;  and  in  1903  it  was  24.6  per  cent,  in  bracts, 
22.3  per  cent,  in  rays,  and  26.4  per  cent,  in  disk-florets.  This  con- 


Sept.  12 _ 17 _ 22 _ 27  Oct.  2 _ 7 


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Fig.  17. — Schematized  representation  of  the 
changes  in  mean  number  of  parts  in  the  heads  of  the 
1903  population;  each  of  the  parallel  oblique  lines 
represents  the  change  in  a  single  stem;  the  heavy  line 
shows  the  change  in  mean  value  for  the  whole  population. 


i9°4] 


SHULL— PLACE-CONSTANTS  FOR  ASTER 


359 


siderable  change  in  the  constants  of  variable  characters  during  the 
single  season  is  now  generally  recognized,  as  has  been  made  manifest 
in  the  discussions  which  were  roused  by  Ludwig’s  (1901)  interpretation 
of  such  differences  as  indicative  of  the  establishment  of  local  races 
or  petites  especes.  Miss  Lee  (1902)  says  in  her  discussion  of  Ludwig’s 
results  and  conclusions,  “we  require  in  fact  to  know  how  the  means, 
variabilities,  and  correlations  of  the  characters  of  a  plant  change 
(i)  with  its  season  and  (ii)  with  the  influence  of  environment  before 
we  can  formulate  a  test  for  racial  differences,”  and  Pearson  (1903) 
and  other  recent  writers  make  similar  statements. 

While  there  is  thus  a  general  recognition  of  the  changes  which 
may  be  expected  to  take  place  during  a  single  season,  there  is  still  a 
question  as  to  changes  of  variability  from  season  to  season.  This 
is  the  first  investigation  in  which  factors  involved  in  modifying  the 
variable  characters  of  plants  or  animals  have  been  so  completely 
limited  to  the  dissimilarity  of  different  seasons.  Although  a  number 
of  students  have  at  times  found  differences  similar  to  those  presented 
in  this  paper,  their  material  has  been  collected  nearly  always  in  such 
a  way  as  to  allow  of  some  other  interpretation,  and  the  conclusions 
arrived  at  have  in  consequence  usually  assumed  the  absence  of 
seasonal  fluctuations. 

Yule  (1902)  has  investigated  the  number  of  sepals  of  Anemone 
nemorosa  growing  in  several  different  habitats  in  the  neighborhood 
of  Bookham,  Surrey,  England,  during  the  years  1889-1900,  but 
unfortunately  his  collections  were  not  made  at  coincident  dates  of 
the  several  years,  and  one  of  the  habitats  had  changed  during  the 
time  in  which  the  observations  were  made  from  an  exposed  clearing 
to  a  well-grown  shady  copse.  Although  he  interprets  his  results  as 
indicating  a  considerable  fluctuation  from  season  to  season,  his  data 
can  be  thrown  into  a  single  series  and  shown  to  exhibit  just  the 
changes  which  recent  investigations  of  Reinohl  (1903)  on  Alsine 
media  and  the  results  recorded  here  for  Aster  prenanthoides  show 
to  occur  during  a  single  season.  Thus,  taking  Yule’s  data  for 
habitat  (C),  which  he  describes  as  a  narrow  strip  of  copse  at  Little 
Bookham,  and  arranging  them  according  to  the  time  of  year  at 
which  each  collection  was  made,  without  regard  to  the  year,  we  have 
for  the  mean  number  of  sepals:  April  8-12,  1899,  6.63;  April  15, 


360 


BOTANICAL  GAZETTE 


[NOVEMBER 


1900,  6.81;  April  21-22,  1898,  6.76;  May  7,  1898,  6.51.  A  com¬ 
parison  of  these  results  with  the  curves  in  fig .  7  will  show  them  to 
be  strictly  comparable  with  the  conditions  exhibited  by  Aster  prenan- 
thoides  in  the  single  season  of  1903.  They  differ,  however,  in  being 
much  less  striking,  the  greatest  change  of  mean  value  in  Anemone 
nemorosa  being  only  4.4  per  cent.,  while  the  greatest  change  in  mean 
value  in  Aster  prenanthoides  was  26.4  per  cent. 

The  Clifton  area  of  Aster  prenanthoides  is  in  a  perfectly  natural 
condition,  and  though  the  region  is  much  visited  for  its  fine  scenery, 
this  particular  spot,  being  less  attractive  to  tourists  and  at  the  same 
time  more  difficult  of  access,  is  not  likely  to  be  at  any  time  seriously 
disturbed.  It  can  be  assumed  with  perfect  assurance  that  there 
were  no  appreciable  differences  in  the  habitat  in  the  two  years  1900 
and  1903,  except  such  as  were  due  to  meteorological  differences,  and 
to  these  factors  or  possibly  to  internal  periodicity,  or  a  combination 
of  these  internal  causes  and  climatic  changes  must  be  attributed  the 
great  differences  found. 

It  has  not  been  infrequent  to  find  great  differences  in  variable 
characters  of  plants  from  markedly  different  habitats,  as  in  the  daisies 
( Chrysanthemum  Leucanthemum  and  C.  segetum)  collected  from  barren 
hills  and  fertile  valleys  by  Ludwig  and  de  Vries.  But  here  at  Clifton, 
Ohio,  in  the  same  spot,  in  the  very  same  group  of  plants,  undoubtedly 
consisting  largely  of  the  uniparental  offspring  of  the  very  same 
individuals,  the  mean  number  of  bracts  was  nearly  12.4  per  cent, 
less  in  1903  than  in  1900,  the  mean  number  of  rays  was  nearly  10 
per  cent,  less,  and  the  mean  number  of  disk-florets  10.6  per  cent.  less. 

If  such  differences  as  these  are  due  to  climatic  fluctuations,  it  is 
of  interest  to  consider  what  factors  may  have  been  important  in 
producing  them.  As  already  mentioned,  Reinohl  (1903)  considers 
the  chief  factor  in  determining  the  number  of  parts  in  the  androecium 
of  Alsine  media  to  be  the  condition  of  the  available  food- supply, 
whether  this  be  dependent  upon  the  character  of  the  soil  or  upon 
photosynthetic  activity  conditioned  by  the  intensity  of  the  light. 
As  the  physical  and  chemical  conditions  of  the  soil  in  the  Clifton 
ravine  were  doubtless  essentially  the  same  in  the  two  years  in  question, 
the  only  soil  factor  which  need  be  taken  into  account  is  water-supply 
as  influenced  by  precipitation.  Reinohl  (1903)  states  that  he  could 


I9°4] 


SHULL— PLACE-CONSTANTS  FOR  ASTER 


36t 


Fig.  18. — Comparison  of  the  climatic  conditions  during  the  growing  season  of 
1900  and  1903;  dotted  lines  for  1900;  unbroken  lines  for  1903;  temperature  curves 
represent  conditions  at  Dayton,  Ohio;  precipitation  curves  are  for  Cedarville,  Ohio; 
and  the  sunshine  curves  for  Cincinnati,  Ohio. 


362 


BOTANICAL  GAZETTE 


[NOVEMBER 


observe  no  influence  produced  by  differences  of  temperature  other 
than  that  of  acceleration  or  retardation,  but  as  conditions  of  nutrition 
are  greatly  affected  by  temperature,  it  is  conceivable  that  it  may  be 
in  some  cases  an  important  factor  in  determining  variability.  On 
these  considerations  I  have  sought  to  compare  the  season  of  1903 
with  that  of  1900  with  respect  to  temperature,  precipitation,  and 
light.  As  the  U.  S.  Weather  Bureau  records  are  not  complete  for 
any  of  these  factors  at  Clifton,  I  have  compared  the  conditions  at 
the  nearest  stations  at  which  complete  records  were  available.  In 
fig.  18  these  comparisons  are  represented  graphically,  the  temperature- 
curves  representing  conditions  at  Dayton,  Ohio,  about  60  km  distant, 
the  precipitation- curves  made  from  data  for  Cedarville,  Ohio,  10  km 
distant,  and  the  curves  for  light-intensity  from  the  self-recording 
instrument  at  Cincinnati,  Ohio,  160 km  distant.  These  data  are 
tabulated  in  Tables  I,  K,  and  L,  along  with  the  eleven-year  or 
twelve-year  normal,  and  such  fragmentary  data  as  were  attainable 
for  Clifton  itself. 

As  this  is  the  first  attempt  to  refer  changes  in  the  variability  of 
plants  in  a  state  of  nature  to  definite  climatic  changes,  there  are 
obvious  difficulties  in  the  way  of  making  satisfactory  interpretations, 
and  these  difficulties  can  be  overcome  only  by  further  study.  We 
need  to  know  (a)  the  relative  importance  of  the  several  factors  involved, 
(b)  the  harmonic  optimum  of  each  climatic  factor  for  the  species  in 
question,  ( c )  whether  the  critical  period  is  that  which  precedes  or 
that  which  accompanies  differentiation,  ( d )  the  time  of  beginning 
and  ending  of  the  period  of  differentiation. 


TABLE  I. 

TEMPERATURE  IN  DEGREES  CENTIGRADE. 


Dayton,  O. 

Clifton,  O. 

1900 

1903 

11-yr.  normal 

1900 

1903 

March . 

2.0 

7-7 

4-3 

8.9 

April . 

11  -3 

10.7 

n-3 

10.3 

10.6 

May . 

18.4 

18.4 

17.4 

18.0 

17.9 

June . 

22.4 

18.9 

22 . 6 

17.9 

July . 

24.7 

24.0 

24.8 

24 -3 

23.0 

August . 

26.2 

23-5 

23.0 

25.1 

22.6 

September . 

22 . 7 

19.7 

19-5 

October . 

17.2 

13 -i 

12.6 

J3- 1 

i9°4] 


SHULL— PLACE-CONSTANTS  FOR  ASTER 


363 


TABLE  K. 

PRECIPITATION  IN  CENTIMETERS. 


Cedarville,  O. 

Clifton,  O. 

1900 

1903 

11-yr.  normal 

1900 

1903 

March . 

6-45 

10.41 

IO.  14 

7.98 

April . 

5-5i 

8.79 

5-87 

4-83 

8.51 

May . 

7.70 

9.07 

8.71 

5.00 

11.58 

Tune . 

6.20 

8.20 

8.18 

11.94 

July . 

9-32 

4. 11 

9-63 

I3-05 

3-°5 

August . 

10.29 

i-75 

5-54 

9.04 

•79 

September . 

2.08 

2.46 

5-74 

October . 

5  • 11 

5-92 

4.67 

5-77 

TABLE  L. 


LIGHT-INTENSITY  AT  CINCINNATI,  OHIO. 


1900 

1903 

12-yr.  normal 

March . 

49$ 

36$ 

45$ 

April . 

68 

42 

56 

May . 

73 

72 

62 

June . 

7i 

62 

72 

July . 

80 

84 

76 

August . 

84 

69 

75 

September . 

76 

81 

72 

October . 

73 

73 

No  study  has  been  made  to  determine  the  period  of  differentiation 
in  Aster  prenanthoides ,  but  I  am  assured  by  Dr.  C.  J.  Chamberlain, 
who  has  studied  Aster  Novae- Angliae,  that  some  of  the  heads  in  that 
species  are  already  blocked  out  by  the  first  of  July.  I  consider  it  a 
fair  assumption  that  the  period  of  differentiation  of  the  parts  of  the 
head  in  this  species  lies  between  June  1  and  August  1. 

If  we  accept  the  normal  climatic  conditions  as  near  the  harmonic 
optima  (and  this  may  not  be  a  very  erroneous  assumption,  since  the 
area  in  question  is  near  the  center  of  range),  we  find  that  the  conditions 
were  more  favorable  in  1900  ( a )  with  respect  to  June  and  July  tem¬ 
peratures,  the  temperature  for  these  months  in  1903  being  consider¬ 
ably  below  normal,  ( b )  in  July  precipitation,  1903  having  less  than 
half  the  normal  precipitation  for  that  month,  and  ( c )  in  light-intensity 
for  every  month,  except  possibly  May,  up  to  August  1,  after  which 
no  factor  could  have  any  further  influence.  It  may  well  be  a  question, 


364 


BOTANICAL  GAZETTE 


[NOVEMBER 


however,  whether  the  harmonic  optimum  for  light-intensity  is  not 
likely  to  be  above  the  normal,  the  shade  habit  of  Aster  prenanthoides, 
as  well  as  of  other  green  shade  plants,  being  assumed  on  account  of 
the  protection  afforded  against  excessive  transpiration,  and  not  against 
excessive  lighting.  If  this  be  true,  the  conditions  in  1900  were  even 
more  favorable  than  here  assumed,  since  with  the  exception  of  July 
the  light-intensity  was  higher  in  1900  than  in  1903,  being  generally 
above  normal  in  the  former  year,  while  in  1 903  it  was  generally  much 
below  normal,  being  strikingly  below  in  April  and  June. 

These  several  advantages  of  1900  over  1903  seem  to  be  offset 
by  the  single  factor  of  precipitation  during  May  and  June,  the  rain¬ 
fall  being  appreciably  below  normal  during  those  two  months  of 
1900.  As  pointed  out  in  the  discussion  of  the  habitat,  it  is  probable 
that  precipitation  is  of  very  slight  importance  in  this  case,  leaving 
the  low  light-intensity  and  low  temperature  of  the  month  of  June, 
1903,  as  probably  the  most  important  factors  in  bringing  about  the 
great  change  in  the  number  of  parts  in  the  heads,  the  factors  of  next 
importance  being  possibly  the  very  high  light-intensity  coupled  with 
slight  precipitation  in  the  month  of  July  1903. 

I  wish  to  repeat  that  these  conclusions  are  based  on  assumptions 
which  need  confirmation.  It  must  not  be  forgotten  that  the  after¬ 
effects  of  a  preceding  season  or  a  rigorous  winter  may  also  be  factors 
of  importance,  or  even  that  there  may  be  an  internal  periodicity 
which  cannot  be  definitely  referred  to  environmental  fluctuations. 

Two  features  of  the  frequency  polygons  for  the  bracts,  rays,  and 
disk-florets  (figs.  8-10)  are  sufficiently  striking  to  warrant  considera¬ 
tion,  their  multimodality  and  their  skewness.  So  much  has  been 
written  upon  the  multimodal  character  of  the  frequency  curves  of 
phyllotactic  organs  that  it  need  only  be  pointed  out  here  that  this 
additional  collection  of  material  shows  no  tendency  to  eliminate  the 
multimodality  observed  in  1900,  and  though  the  errors  of  random 
sampling,  which  are  very  great  in  material  of  such  wide  range,  must 
be  held  to  account  for  most,  if  not  all,  of  the  irregularities  of  these 
curves  (Pearson  1902),  there  are  some  evidences  that  permanent 
modes  may  be  developed  on  the  Fibonacci  series  and  Ludwig’s 
“Unterzahlen.” 

The  constant  recurrence  of  this  series  is  not  to  be  taken,  however, 


i9°4] 


SHULL— PLACE-CONSTANTS  FOR  ASTER 


365 


as  has  been  maintained  by  Ludwig  (1899,  1901),  as  proof  that 
variation  in  plants  is  fundamentally  different  from  that  in  animals. 
When  the  phyllotactic  series  shall  have  been  successfully  analyzed, 
they  may  be  found  to  result  from  the  working  out  of  more  or  less 
definite  cell-lineages  as  supposed  by  Ludwig  (1888),  or  they  may  be 
the  result  of  purely  mechanical  relations,  as  believed  by  Schwen- 
dener,  followed  by  Weisse  (1897)  and  Church  (1904),  but  either 
hypothesis,  in  explaining  the  occurrence  of  such  series,  must  leave 
departures  from  the  theoretical  numbers  to  be  accounted  for  as 
fluctuating  variations.  In  addition  to  this  variation  about  each  num¬ 
ber  of  the  series,  there  is  the  general  variation  which  may  have  a 
sufficiently  wide  range  to  allow  the  variates  to  coincide  with  two  or 
more  numbers  of  the  phyllotactic  series,  so  that  we  have  in  the  case 
of  phyllotactic  variants  two  series  of  variations,  the  one  overlying  and 
partially  masking  the  other.  There  can  be  little  doubt  that  these 
variations  taken  separately  will  be  found  to  agree  with  all  the  laws  of 
variation  determined  for  animals  and  the  non-phyllotactic  characters 
of  plants. 

Although  de  Vries  (1899&)  was  able  by  selection  to  establish 
races  of  Chrysanthemum  segetum  having  monomodal  ray-curves,  this 
must  not  be  taken  as  supporting  Ludwig’s  (1901)  view  that  multi¬ 
modality  is  due  to  the  establishment  of  a  mixed  population  of  petites 
especes  through  the  common  occurrence  of  asexual  and  autogamic 
sexual  reproduction,  for  Reinohl  (1903)  was  able  to  reduce  the 
multimodal  curves  of  Alsine  media  to  monodal  curves  without  selec¬ 
tion ,  by  different  degrees  of  light  and  manuring. 

It  is  to  be  hoped  that  we  shall  soon  have  a  method  of  treatment  of 
phyllotactic  variants  which  will  remove  the  Fibonacci  mask  and 
permit  the  analysis  of  the  underlying  individual  variation  with  as 
much  precision  as  is  now  attained  with  non-phyllotactic  variants. 

Although  it  is  impossible  on  account  of  the  multimodality  of  these 
curves  to  analyze  the  skewness,  it  is  so  marked  in  the  case  of  the  bracts 
and  rays  (figs.  8  and  p)  as  to  be  recognized  at  a  glance.  There  have 
been  various  interpretations  of  skewness  in  different  connections, 
favorite  early  views  (Davenport  1901)  being  that  it  results  either 
by  the  elimination  of  one  or  other  of  the  extremes  through  the 
process  of  natural  selection,  or  that  heterogeneity  is  introduced  by  the 


366 


BOTANICAL  GAZETTE 


[NOVEMBER 


development  of  a  new  race  within  the  range  of  the  old  but  centering 
about  a  different  mean.  It  is  also  believed  that  skewness  may  result 
from  physiological  causes  having  no  direct  bearing  upon  the  origin 
or  modification  of  species.  While  in  no  specific  case  may  the 
suggested  interpretation  be  the  correct  one,  these  different  views  may 
at  least  be  accepted  as  evidence  that  skewness  may  result  from  various 
causes,  and  that  it  is  therefore  not  self-explanatory. 

If  the  1903  curves  are  compared  with  those  for  1900  in  figs.  11-13, 
it  will  be  seen  that  in  every  case  the  positive  sides  of  the  curves  are 
approximately  coincident,  but  on  the  negative  side  there  is  a  very 
material  disagreement.  According  to  the  recent  discussion  of  skew 
variation  by  Lutz  (1904),  we  have  here  a  case  of  skewness  produced 
by  the  addition  of  variates,  and  this  addition  of  such  magnitude  as 
already  to  overtop  the  1900  population,  thus  giving  a  fine  example  of 
“historic”  skewness;  but  no  one  can  be  convinced  that  this  is  here 
due  to  the  “starting  of  a  new  race  about  a  mean  within  the  range 
of  the  old  race.” 

It  is  evident  that  the  skewness  is  here  the  result  of  direct  physio¬ 
logical  reaction  to  the  changed  environment.  Not  all  individuals 
are  alike  sensitive  to  changed  conditions,  some  being  more,  some  less 
affected  by  a  given  amount  of  change ;  so  that  while  many  individuals 
respond  to  the  less  favorable  conditions  by  the  production  of  heads 
with  smaller  numbers  of  parts,  there  is  still  a  considerable  number 
of  conservative  individuals  which  are  little  or  not  at  all  affected.  The 
positive  skewness  of  these  curves  is  due  to  the  fact  that  on  y  a  small 
proportion  of  the  population  is  conservative.  If  the  great  mass  of 
variates  had  been  comparatively  conservative  and  only  a  small  per¬ 
centage  sensitive  to  the  changed  conditions,  it  is  plain  that  ,he  posi¬ 
tion  of  the  principal  modes  would  have  been  little  affected,  while  the 
mean  would  have  been  lowered  and  negative  skewness  would  have 
been  the  result.  This  would  then  have  been  a  case  of  so-called 
“prophetic”  skewness.  We  may  say  then  that  in  cases  of  direct  or 
physiological  variation,  prophetic  skewness  indicates  slight  sensitive¬ 
ness,  and  historic  skewness  great  sensitiveness,5  to  the  changed  con¬ 
ditions,  provided  always,  of  course,  that  under  ordinary  condit'ons 
the  distribution  of  the  variates  affected  is  normal. 

s  As  measured  by  the  number  of  sensitive  individuals,  not  by  the  degree  of  sen¬ 
sitiveness  of  each  individual. 


i9°4] 


SHULL— PLACE-CONSTANTS  FOR  ASTER 


367 


Cases  are  well  known  in  which  the  distribution  does  not  appear 
to  be  normal  under  any  ordinary  conditions,  the  frequency  curves 
being  of  the  “half  Galton”  type,  as  for  instance  the  petals  of  Caltha 
palustris,  Potentilla  anserina ,  Ranunculus  bulbosus  (de  Vries  1894), 
Ranunculus  repens  (Pledge  1897),  sepals  and  petals  of  Ranunculus 
arvensis  (Burkill  1902),  leaflets  of  clover  (de  Vries  1899a),  ascidia 
and  other  abnormalities  of  various  species  (de  Vries  1899a,  Tammes 
1903),  and  other  characters.  Such  cases  may  not  be  really  so 
exceptional,  however,  as  they  at  first  appear.  We  have  only  to 
assume  that  the  normal  condition  for  these  characters  is  one  in  which 
the  value  of  cr  approaches  zero  to  see  that  these  are  cases  of  “pro¬ 
phetic”  skewness  due  to  the  small  proportion  of  abmodal  variates; 
in  other  words,  due  to  slight  sensitiveness  to  conditions  tending  to 
produce  a  number  of  organs  higher  or  lower  than  the  normal  mode. 

It  may  be  found  that  any  population  or  even  any  species  is  suffi¬ 
ciently  uniform  in  its  reactions  to  various  degrees  of  environmental 
change  to  allow  us  to  derive  from  the  direction  and  amount  of  skew¬ 
ness  the  approximate  value  of  the  mean  under  average  conditions 
or  under  conditions  which  would  give  a  normal  distribution  of  the 
variates.  Thus,  the  knowledge  that  this  population  of  Aster  prenan- 
thoides  is  so  sensitive  to  change  as  to  exhibit  strong  positive  skewness 
when  conditions  are  below  average  may  be  found  to  warrant  the 
assumption  that  there  will  be  a  strong  negative  skewness  under 
unusually  favorable  conditions,  and  also  that  the  skewness  exhibited 
by  a  collection  from  any  new  locality  would  give  an  indication  by 
its  direction  as  to  whether  that  collection  was  below  or  above  the 
average  prevailing  condition  for  that  place.  But  before  we  can 
apply  this  principle  with  any  confidence  in  determining  the  “normal 
mean”  of  any  particular  population,  it  will  be  necessary  to  confirm 
our  assumptions  (a)  that  the  distribution  for.  that  population  is  normal 
under  average  conditions,  and  ( b )  that  the  sensitiveness  to  unusually 
favorable  conditions  is  similar  in  intensity  to  the  sensitiveness  to 
unfavorable  conditions. 

The  principle  here  presented  of  variability  in  individual  sensitive¬ 
ness  to  changes  of  environment  is  likely  to  find  a  wide  applicability 
in  the  interpretation  of  skew  variation,  and  suggests  the  need  of  first 
determining  whether  or  not  there  is  direct  variation  of  the  organ  or 


368 


BOTANICAL  GAZETTE 


[NOVEMBER 


character  under  consideration  before  assuming  that  either  natural 
selection  or  mutation  is  involved  in  any  given  case  of  skewness.  And 
although  this  is  most  strikingly  true  of  plants,  it  must  likewise  be 
true  of  animals,  especially  of  animals  having  a  short  life-cycle,  so 
that  no  investigation  can  be  considered  as  giving  satisfactory  support 
to  any  hypothesis  of  evolution  until  the  sensitiveness  of  the  character 
under  consideration  to  secular  changes  shall  have  been  determined. 

Perhaps  even  more  remarkable  than  the  skewness  and  the  changes 
in  mean  value,  which  have  resulted  from  the  less  favorable  conditions 
in  1903,  is  the  great  increase  in  value  of  the  coefficient  of  variability. 
Reference  to  Tables  E,  F,  and  G  will  show  that  the  variability  in 
the  bracts  in  1900  was  12. 979 ±.241,  as  compared  with  19. 928 ±.345 
in  1903.  Corresponding  changes  are  shown  in  rays  and  disk- florets, 
from  14.516i.270  to  19. 766  +  . 343,  and  from  12.546i.233  to  21.595 
i.374,  respectively.  As  it  has  been  assumed  that  the  low  mean 
values  indicate  that  conditions  were  less  favorable  in  1903  than  in 
1900,  we  may  accept  these  changes  in  the  coefficients  of  variability 
as  proof  of  the  hypothesis  that  when  organisms  are  introduced  into 
unusual  surroundings  or  subjected  to  unusual  conditions  they  become 
more  variable,  and  that  this  would  be  favorable  to  any  selective 
process  which  might  set  in  as  a  result  of  the  change.  Before  too 
great  stress  is  laid  on  this  conclusion,  however,  we  need  to  consider 
the  nature  of  the  coefficient  of  variability.  The  importance  of  this 
constant  lies  in  the  fact  that  it  is  an  abstract  number  and  therefore 
allows  us  to  compare  the  variability  in  characters  of  different  magni¬ 
tude  or  even  of  different  quality,  as  color,  form,  size,  weight,  number, 
etc.  It  consists  of  two  factors,  the  standard  deviation  (<r)  and  the 

mean  (M),  and  is  expressed  by  the  formula  C.  V.  =  °°  .  The 

value  of  the  coefficient  of  variability  will  change  directly  with  changes 
of  cr  and  inversely  with  changes  of  the  mean.  Turning  now  to  the 
cause  of  the  greatly  increased  coefficient  of  variability,  we  find  upon 
inspecting  Tables  E,  F,  and  G  that  the  value  of  cr  was  in  every  case 
considerably  higher  in  1 903  than  in  1 900,  and  at  the  same  time  that 
the  mean  was  much  lower,  so  that  both  factors  acted  together  in 
producing  the  high  values  of  the  coefficient  of  variability. 

To  show  that  this  coefficient  is  not  always  a  satisfactory  measure 


I9°4] 


SHULL— PLACE-CONSTANTS  FOR  ASTER 


369 


of  variability,  let  us  assume  that  conditions  had  been  unusually 
favorable  to  such  a  degree  as  to  give  curves  with  the  same  values  of 
cr j  but  negatively  skew.  The  variability  would  then  be  approximately 
the  same,  but,  instead  of  the  coefficient  being  the  same  or  even 
nearly  the  same,  it  would  be  very  much  less,  owing  to  the  greatly 

IOO  cr 

increased  value  of  the  mean.  I  do  not  think  that  — —  gives  a  proper 

value  of  the  coefficient  of  variability  in  cases  of  skew  variation,  since 
its  values  in  positively  skew  curves  are  not  comparable  with  those 
in  curves  of  the  same  species  or  even  of  the  same  population,  which 
are  negatively  skew.  If  the  “normal  mean”  could  be  derived  from 
skew  curves,  that  might  be  used  instead  of  the  mean  in  the  formula 
for  the  coefficient  of  variability,  thus  making  the  value  of  cr  alone 
indicate  the  changes  of  variability  from  time  to  time  within  one  and 
the  same  population.  This  would  be  theoretically  correct,  but  it 
must  be  evident  that  the  experimental  determination  of  the  normal 
mean,  except  through  a  long  series  of  investigations  upon  any  popula¬ 
tion  under  consideration,  is  impossible,  even  though,  as  pointed  out 
above,  the  degree  and  direction  of  skewness  may  in  some  cases  give 
a  rough  approximation  to  it  when  the  sensitiveness  of  the  species 
in  question  is  known. 

Returning  now  to  the  question  as  to  the  increased  variability  due 
to  changed  environmental  conditions,  we  find  that  the  present  imper¬ 
fect  coefficient  of  variability,  which  would  tend  to  minimize  the 
variability  when  conditions  are  unusually  favorable,  would  still  be 
considerably  increased  by  such  unusually  favorable  conditions  as 
would  result  in  a  negative  skewness  equal  in  magnitude  to  the  positive 
skewness  of  the  1903  curves.  We  may  confidently  accept  the  results 
/  of  this  study  as  proof,  therefore,  that  changes  of  environment  do 
result  in  increased  variability. 

It  was  noted  in  1900  that  the  correlations  between  the  parts  in  the 
head  were  very  high,  and  by  reference  to  Table  H  it  will  be  seen  that 
in  1903  they  were  very  considerably  higher  still,  the  highest  coefficient 
in  both  years  being  that  between  bracts  and  rays.  The  exact  meaning 
of  changes  in  the  degree  of  organic  correlation  is  proving  a  somewhat 
puzzling  problem  at  the  present  time.  Ludwig  (1901)  presents  a 
striking  case  of  this  kind  as  evidence  of  racial  distinctness  between 


37° 


BOTANICAL  GAZETTE 


[NOVEMBER 


two  populations  of  Ranunculus  jicaria ,  but  MacLeod  (1899)  has 
shown  that  sim'lar  changes  may  be  found  in  that  spec'es  at  d  fferent 
times  in  a  single  season.  I  have  also  found  (Shull  1902)  that  the 
coefficients  of  correlation  in  Aster  prenanthoides  may  be  very  different 
at  different  parts  of  the  season. 

Before  the  significance  of  such  changes  can  be  understood  it  will 
be  necessary  to  investigate  the  nature  of  correlation  when  considered 
in  this  statistical  way.  Some  biologists  use  the  term  “correlation” 
to  designate  a  relation  between  two  organs  or  characters,  such  that 
the  development  of  the  one  determines  that  of  the  other,  as  for  instance 
the  dependence  of  the  secondary  sexual  characters  upon  the  primary 
in  animals,  or  the  relation  of  the  internodes  to  the  leaves  in  plants. 
In  this  kind  of  correlation  the  failure  of  the  one  organ  or  character 
to  develop,  or  its  removal  at  an  early  stage  of  development,  invariably 
prevents  or  modifies  the  development  of  the  other.  Every  degree  of 
correlation  in  this  sense  is  found  in  different  cases,  and  it  probably 
exists  to  some  extent  even  between  organs  whose  immediate  relations 
to  each  other  are  little  understood.  It  is  only  rarely,  however,  that 
this  kind  of  correlation  is  not  insignificant  as  compared  with  biometri¬ 
cal  correlation.  Thus,  in  the  biometrical  sense  there  is  a  very  high 
correlation  between  the  index  fingers  of  the  right  and  left  hands, 
but  the  removal  of  one  of  these  would  have  no  appreciable  effect 
upon  the  development  of  the  other. 

For  convenience  we  may  speak  of  “immediate”  or  “direct” 
correlation  when  one  organ  or  character  stands  in  a  d  rect  causal 
relation  to  another,  and  “mediate”  or  “indirect”  correlation  in  cases 
of  correlated  variation  in  which  no  such  direct  dependence  exists. 
Statist' cal  measures  of  correlation  make  no  distinction  between  these 
two  kinds  of  correlation,  but  as  a  notable  degree  of  immediate  corre¬ 
lation  is  comparatively  rare,  while  mediate  correlation  is  almost 
universal,  the  correlation  of  parts  as  spoken  of  by  the  biometrician 
may  be  considered  as  mediate  or  indirect.  Mediate  correlation 
between  two  organs  or  characters  may  be  defined,  then,  as  their 
mutual  relation  to  the  combination  of  common  causes,  such  as  hered¬ 
ity,  nutrition,  etc.,  which  determine  their  quantitative  relations. 
It  is  the  relation  which  results  in  proportion  and  symmetry.  When 
mediate  correlation  is  perfect,  i.  e.,  when  p  =  1,  the  two  organs  or 


igc4]  SHULL— PLACE-CONSTANTS  FOR  ASTER  371 

characters  are  proportionately  influenced  by  every  variation  in  the 
factors  which  determine  their  size,  number,  or  other  quantitative 
relation,  and  neither  is  affected  by  any  factor  which  does  not  affect 
the  other.  The  organs  do  not  modify  each  other,  but  both  are 
affected  by  the  same  conditions.  Only  confusion  results  from  the 
failure  to  appreciate  the  difference  between  immediate  and  mediate 
correlation,  as  may  be  seen  in  Burkill’s  (1902)  discussion  of  the 
correlation  in  the  parts  of  the  flower  of  Ranunculus  arvensis,  when 
he  says  that  “reduction  in  the  number  of  petals  does  not  act  as  a 
reflex  on  the  number  of  sepals  in  anything  like  the  way  in  which 
the  reduction  of  sepals  may  be  said  to  promote  reduction  of  petals.” 

If  as  the  values  of  any  pair  of  mediately  correlated  organs  or 
characters  are  increased  or  decreased  the  correlation  between  them 
is  changed,  it  must  mean  that  one  or  other  of  them  becomes  propor¬ 
tionately  less  sensitive  to  the  causes  producing  the  change  of  values, 
and  becomes  more  fixed  or  more  variable  in  its  quantitative  relations. 
Such  a  change  is  well  illustrated  by  an  interesting  diagram  presented 
by  Burkill  (1902),  in  which  it  is  shown  that  sepals,  petals,  stamens, 
and  carpels  of  Ranunculus  arvensis  vary  together,  i.  e.,  are  closely 
correlated,  in  flowers  having  the  total  number  of  parts  less  than  19, 
but  in  flowers  having  a  higher  total  number  of  parts  the  sepals  become 
fixed  in  number  at  5,  and  the  correlation  between  sepals  and  the 
parts  which  continue  to  increase  becomes  zero.  In  flowers  with 
more  than  22  parts  the  mean  number  of  petals  likewise  becomes 
fixed  at  5.  In  flowers  of  still  higher  numbers  of  parts  the  carpels 
show  a  tendency  to  respond  with  proportionately  less  increase  as 
compared  with  the  stamens.  It  is  plain  then  that  in  this  species 
any  conditions  which  promote  the  formation  of  flowers  with  a  high 
number  of  parts  will  tend  to  decrease  the  degree  of  correlation  and 
vice  versa. 

But  it  is  an  important  fact  which  must  not  be  overlooked  that 
changes  in  the  coefficient  of  correlation  do  not  necessarily  mean  an 
actual  change  in  correlation.  Pearson  (1903)  has  pointed  out  that 
heterogeneity  in  a  population  tends  to  increase  the  coefficient  of 
correlation,  but  of  course  such  heterogeneity  does  not  increase  the 
actual  degree  of  correlation.  It  is  probable  that  most  of  the  marked 
changes  which  have  thus  far  been  observed  in  coefficients  of  correla- 


372 


BOTANICAL  GAZETTE 


[NOVEMBER 


tion  are  to  be  accounted  for  in  this  way.  I  have  already  shown  that 
my  first  collection  n  1900  was  made  long  after  the  beginning  of  the 
flowering  season,  and  hence  had  the  earliest  heads  with  low  numbers 
of  parts  associated  with  the  heads  having  the  highest  numbers  of  parts 
produced  during  the  season,  and  this  fact  sufficiently  explains  the 
high  correlations  found  in  that  collection.  A  similar  explanation 
may  account  for  the  considerable  increase  in  the  coefficients-  of  corre¬ 
lation  between  the  parts  of  the  heads  in  1903  as  compared  with  those 
of  1900,  as  there  are  associated  in  the  1903  collection  the  heads  of 
conservative  individuals  and  those  of  individuals  which  were  much 
modified  because  of  their  great  sensitiveness  to  the  unfavorable 
conditions  in  the  latter  year.  It  is  apparent,  therefore,  that  in  cases 
of  changed  coefficients  of  correlation,  as  in  other  cases,  it  is  necessary 
to  scrutinize  carefully  the  influence  of  more  or  less  artificial  conditions 
upon  the  value  of  the  constants  before  we  can  appreciate  their  biological 
significance 

The  results  of  ,his  study  have  fully  borne  out  the  suggestion  that 
considerable  differences  may  occur  in  individual  variation  from 
year  to  year,  and  it  shows  that  such  differences  may  be  even  greater 
than  one  would  expect.  It  is  not  likely  that  this  is  an  extreme  case, 
nor  that  the  differences  between  these  two  collections  is  even  near 
the  limit  for  this  species.  To  some  these  results  may  seem  to  pre¬ 
clude  the  possibility  of  deriving  anything  of  further  value  from  quan¬ 
titative  studies  of  variation,  while  to  others  many  new  problems  of 
great  interest  and  importance  will  be  suggested.  The  interpretations 
which  students  have  based  upon  the  assumption  that  seasonal  fluc¬ 
tuations  do  not  occur  will  have  to  be  greatly  revised  or  discarded 
altogether,  and  before  we  can  appreciate  the  exact  bearing  of  any 
case  of  variation  upon  the  great  problems  of  evolution  it  will  be  neces¬ 
sary  to  know  the  laws  governing  that  variation.  It  is  to  problems  of 
this  nature  that  students  must  direct  their  earnest  attention  if  we 
are  ever  to  have  a  basis  for  the  appreciation  of  the  bearing  of  indi¬ 
vidual  variation. 

VI.  SUMMARY. 

A  second  collection  of  heads  of  Aster  prenanthoides .  Muhl.  was 
made  in  1903  from  the  same  area  at  Clifton,  Ohio,  that  supplied 
material  for  a  quantitative  study  in  1900.  The  bracts,  rays,  and 


i9°4] 


SHULL— PLACE-CONSTANTS  FOR  ASTER 


373 


disk-florets  were  studied  quantitatively,  and  the  results  compared 
with  those  of  the  earlier  study. 

Twelve  successive  collections  were  made  from  the  same  plot,  and 
it  was  found  that  the  earliest  collection  had  low  mean  numbers,  that 
the  mean  values  then  leaped  quickly  to  a  maximum,  falling  gradually 
to  near  the  end  of  the  season,  and  that  the  last  collection  exhibited  a 
rise,  the  rise  in  mean  values  at  the  beginning  and  at  the  end  of  the 
season  being  in  disagreement  with  the  conclusion  reached  in  my 
earlier  study.  In  general,  the  first  head  to  bloom  on  any  stem  has 
the  highest  number  of  parts  possessed  by  any  head  produced  by  that 
stem,  and  the  last  to  bloom  has  the  lowest  number.  The  low  mean 
numbers  at  the  beginning  of  the  season  are  due  to  the  precocious 
flowering  of  the  weakest  individuals,  and  similarly  the  rise  at  the  end 
of  the  season  is  due  to  the  belated  flowering  of  a  few  very  vigorous 
individuals. 

Comparison  of  the  results  with  those  of  1900  show  that  the  mean 
values  in  1903  were  10-12  per  cent,  lower  than  in  1900,  ^nd  that 
accompanying  these  low  mean  values  there  are  a  strong  positive 
skewing  of  the  curves,  a  remarkable  rise  in  the  coefficient  of  varia¬ 
bility,  and  a  considerable  increase  in  the  coefficient  of  correlation. 

The  difference  in  the  mean  values  for  the  two  years  is  attributed 
to  less  favorable  climatic  conditions  in  1903,  chiefly  to  low  tempera¬ 
ture  and  low  light-intensity  in  the  month  of  June. 

The  skewness  is  due  to  the  unequal  sensitiveness  of  individuals  to 
changes  of  environment.  It  is  positive  because  the  proportion  of 
conservative  individuals  is  small  In  direct  or  physiological  variation, 
“historic”  skewness  indicates  great  sensitiveness  and  “prophetic” 
skewness  indicates  slight  sensitiveness  to  the  changes  of  environment. 

The  great  increase  in  the  coefficient  of  variability  is  due  to  an 
increase  in  the  standard  deviation  and  a  decrease  of  the  mean.  The 
present  coefficient  of  variability  is  not  satisfactory  in  cases  of  skew 
variation,  and  the  value  of  cr  alone  should  be  used  as  the  measure  of 
changes  of  variability  in  one  and  the  same  population. 

Changes  in  the  coefficient  of  correlation  may  be  due  either  to  an 
actua1  change  of  correlation  or  to  the  introduction  of  a  greater  or  less 
degree  of  heterogeneity.  The  latter  is  probably  responsible  for  the 
changes  noted  in  this  species 


374 


BOTANICAL  GAZETTE 


[NOVEMBER 


I  gratefuky  acknowledge  my  indebtedness  to  Dr.  C.  B.  Daven¬ 
port,  under  the  inspiration  of  whose  lectures  this  study  was  under¬ 
taken  and  under  whose  direction  it  was  largely  carried  on;  to  M'ss 
Olive  D.  Coe  for  the  care  with  which  the  material  was  collected, 
for  the  negatives  from  which  figs,  i,  4,  and  5  were  reproduced,  and 
for  the  original  of  fig.  6;  to  the  Directors  of  the  U.  S.  Weather  Bureau 
stations  at  Columbus,  Ohio,  and  at  Cincinnati,  Ohio,  for  climato¬ 
logical  data;  and  to  the  curators  of  numerous  public  and  private 
herbaria  for  the  data  upon  which  fig.  j  is  based. 

Station  for  Experimental  Evolution, 

Cold  Spring  Harbor,  Long  Island,  N.  Y. 


LITERATURE  CITED. 

Adams,  C.  C.  1902.  Southeastern  United  States  as  a  center  of  geographica 
distribution  of  flora  and  fauna.  Biological  Bull.  3:115-131. 

Burkill,  I.  H.  1895.  On  some  variations  in  the  number  of  stamens  and  carpels. 
Jour.  Linn.  Soc.  Bot.  31:216-245. 

- .  1902.  On  the  variation  of  the  flower  of  Ranunculus  arvensis.  Jour. 

Asiatic  Soc.  Bengal  71:93-120. 

Church,  A.  H.  1904.  The  principles  of  phyllotaxis.  Ann.  Botany  18: 227-243. 
Davenport,  C.  B.  1899a.  The  importance  of  establishing  specific  place-modes. 
Science  N.  S.  9:415-416. 

- .  18996.  Statistical  methods  with  special  reference  to  biological  varia¬ 
tion.  New  York:  John  Wiley  &  Sons.  2d  ed.,  revised  and  enlarged.  1904. 

- .  1901.  Zoology  of  the  twentieth  century.  Science  N.  S.  14:315-324. 

Haacke,  W.  1896.  Ueber  numerische  Variation  typischer  Organe  und  kor- 
relative  Mosaikarbeit.  Biol.  Centralbl.  16:481-497,  529-547. 

Lee,  Miss  Alice.  1902.  Dr.  Ludwig  on  variation  and  correlation  in  plants. 
Biometrika  1:316-319. 

Ludwig,  F.  1888.  Weitere  Kapitel  zur  mathematischen  Botanik.  V.  Die  Zell- 
theilung  und  der  gesetzmassige  Aufbau  der  Bacillarienbander.  VI.  Das 
Vorkommen  bestimmter  Zahlen  bei  den  Organen  hoherer  Gewachse  und  das 
Vermehrungsgesetz  des  Fibonacci.  Zeitschr.  f.  math.  u.  naturwiss.  Unterr. 
I9:32I-33^- 

- .  1899.  Ben  fondamenteel  verschil  in  de  veranderlijkheid  bij  het  dier 

en  de  planten  ?  Kruidkundig  Genootschap  Dodonaea  te  Gent  11 : 108-121. 

- .  1901.  Variationsstatistische  Probleme  und  Materialien.  Biometrika 

1 : 11-29. 

Lutz,  F.  E.  1904.  Biological  interpretation  of  sk.ew  variation.  Science  N.  S. 
19:214. 


I9°4] 


SHULL— PLACE-CONSTANTS  FOR  ASTER 


37f~ 


MacLeod,  J.  1899.  Over  de  correlatie  tusschen  het  aantal  meeldraden  en  het 
aantal  stampers  bij  het  Speenkruid  ( Ficaria  ranunculoides).  Bot.  Jaarboek 

11 : - Discussed  by  F.  R.  Weldon  in  Biometrika  1:125-128. 

Pearson,  K.  1902.  On  the  sources  of  apparent  polymorphism  in  plants,  etc. 
Biometrika  1:304-306. 

- .  1903.  Variation  and  correlation  in  the  lesser  celandine  from  diverse 

localities.  Biometrika  2 : 145-164. 

Pledge,  J.  H.  1897.  Numerical  variation  of  parts  of  Ranunculus  repens.  Nat. 
Sci.  10:323-328. 

Reinohl,  F.  1903.  Die  Variation  im  Androecium  der  Stellaria  media  Cyr. 
Bot.  Zeit.  61:159-200. 

Shull,  G.  H.  1902.  A  quantitative  study  of  variation  in  the  bracts,  rays,  and 
disk-florets  of  Aster  Shortii  Hook.,  A.  Novae- Angliae  L.,  A.  puniceus  L.,  and 
A.  prenanthoides  Muhl.,  from  Yellow  Springs,  Ohio.  Amer.  Nat.  36:111- 
152. 

Smallwood,  Miss  Mabel  E.  1903.  The  beach  flea:  Talorchestia  longicornis. 
Cold  Spring  Harbor  Monographs  I.  Brooklyn:  The  Brooklyn  Institute  of 
Arts  and  Sciences. 

Tammes,  Fraulein  Tine.  1903.  Die  Periodicitat  morphologischer  Erschei- 
nungen  bei  den  Pflanzen.  Verhandl.  Kon.  Akad.  Wetenschappen  te  Amster¬ 
dam.  Amsterdam:  Johannes  Muller. 

Tower,  W.  L.  1902.  Variation  in  the  ray-flowers  of  Chrysanthemum  Leucan- 
themum  L.  at  Yellow  Springs,  Greene  co.,  Ohio,  with  remarks  upon  the 
determination  of  modes.  Biometrika  1:309-315. 

Vries,  H.  de.  1894.  Ueber  halbe  Galton-Curven  als  Zeichen  discontinuirlicher 
Variation.  Ber.  Deutsch.  Bot.  Gesells.  12:197-207. 

- .  1899a.  Ueber  die  Periodicitat  der  partiellen  Variationen.  Ber. 

Deutsch.  Bot.  Gesells.  17:45-51. 

- .  1899 b.  Ueber  Curvenselection  bei  Chrysanthemum  segetum.  Ber. 

Deutsch.  Bot.  Gesells.  17:84-98. 

Weisse,  A.  1897.  Die  Zahl  der  Randbliithen  am  Compositenkopfchen.  Jahrb. 
Wiss.  Bot.  30:453-483.  pi.  ig. 

Yule,  G.  U.  1897.  On  the  theory  of  correlation.  Jour.  Roy.  Statistical  Soc. 
60 : 44.  pt.  4. 

- .  1902.  Variation  of  the  number  of  sepals  in  Anemone  nemorosa.  Bio¬ 
metrika  1:307-309. 


